If \(4(x+2 y)=8 y\), then \(y\) is: A. \(4 \mathrm{x}\) B. \(4 x+4 y\) C. 1 D. 0 E. Cannot be determined

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It's important because it forms the foundation for most higher mathematics and many real-life problems.
In this exercise, we start with an equation involving variables and constants: \(4(x + 2y) = 8y\).

• The variable \(x\) represents an unknown quantity we want to solve for.
• Similarly, \(y\) represents another unknown quantity.

At its core, algebra aims to find the values of these unknowns that make the equation true. Understanding how to manipulate and simplify these expressions is crucial, which leads us to the next concept.

Simplifying an equation means to manipulate it to make it simpler and more straightforward to solve. This usually involves expanding, combining like terms, and eliminating variables.

Let's walk through the simplification process for our given equation.

• First, we expand the left side by distributing the 4 across \(x + 2y\). This gives us: \(4x + 8y = 8y\).
• Next, we want to get all terms involving the same variable on one side of the equation. We do this by subtracting \(8y\) from both sides: \(4x + 8y - 8y = 8y - 8y\).
• This simplifies to: \(4x = 0\).
• Finally, we solve for \(x\) by dividing both sides by 4: \(x = 0\).

Each step in this process brings us closer to isolating the variable we need to solve, making the problem more manageable.

Mathematical reasoning involves logic and critical thinking to solve problems. It's not just about following steps, but also understanding why those steps work.
In our example, once we've simplified to find that \(x = 0\), the next logical step is to consider what this tells us about \(y\).
Let's remember:

• We started with \(4(x + 2y) = 8y\).
• After simplifying, we found \(4x = 0\).
• This implies \(x = 0\).

With \(x = 0\), our original equation to find \(y\) simplifies to a straightforward solution. By logically following the steps and understanding the relationship between the variables, we conclude that \(y\) must be zero (D. 0). Finding this final value demonstrates how key concepts in algebra, equation simplification, and reasoning work together to solve complex problems.